%\documentclass[Atmos. Chem. Phys.]{copernicus}
\documentclass[Atmos. Chem. Phys.]{copernicus_discussions}
%\usepackage{color}
%\usepackage{lineno}
\usepackage{longtable}

\usepackage{amssymb,amsmath} 

\begin{document}\sloppy

% potential titiles 
%  
% 1. A global estimate of ion production rate in the lower troposphere induced by radioactive decay of radon and its daughters
% 2. Sensitivity of simulated radon activity in the lower troposphere to using different radon fluxes: a GCM study
% 3. Simulation of radon transport and ion production by decay of radon in the lower troposphere
%
% final title 

\title{Radon decay formula} 

\author[1]{K.~Zhang}  

\affil[1]{Max Planck Institute for Meteorology, Hamburg, Germany} 

\runningtitle{radon and ion production}
\runningauthor{Zhang et al.}

\correspondence{K.~Zhang (kai.zhang@zmaw.de)}


%\received{01 July 2010}
%\accepted{31 December 2010}
%\published{21 Jan 2011}

% radon unit converstion
%  
% 1 Ci  = 3.7E+10 Bq 
% 1 Bq  = 1000 mBq 
% 1 pCI = 56 mBq 
%  


\firstpage{1}
\maketitle

\begin{abstract}
Formula for radon decay series. 
\end{abstract}


\introduction
 


\begin{equation}
  A = \alpha e^{-\lambda_{1}t}
  \int_t^{t+\Delta t} \! A \, dt = \alpha e^{-\lambda_{1}t}
\label{eqn:A}
\end{equation}


\begin{equation} 
  \int_t^{t+\Delta t} \! A \, dt = \int_t^{t+\Delta t} \! \alpha e^{-\lambda_{1}t} \, dt
                                 = \alpha [e^{-\lambda_{1}t} - e^{-\lambda_{1}t+\Delta t} ] / \lambda_{1}
\label{eqn:A}
\end{equation}


\begin{equation} 
  \int_t^{t+\Delta t} \! B \, dt 
= \int_t^{t+\Delta t} \! 
  [ \beta e^{-\lambda_{2}t} + 
    \alpha \lambda_{1} (e^{-\lambda_{1}t} - e^{-\lambda_{2}t}) / (\lambda_{2} - \lambda_{1}) 
  ] \, dt
\label{eqn:B}
\end{equation}



\begin{equation} 
  \int_t^{t+\Delta t} \! C \, dt 
= \int_t^{t+\Delta t} \! 
  [ \gamma e^{-\lambda_{3}t} + \\ 
    \frac{ \alpha \lambda_{2} \lambda_{1} } { (\lambda_{3} - \lambda_{1}) (\lambda_{2} - \lambda_{1}) } 
    (e^{-\lambda_{1}t} - e^{-\lambda_{3}t}) + \\ 
    \frac{\lambda_{2}} {(\lambda_{3} - \lambda_{2})} 
    (\beta - \frac{\alpha \lambda_{1}} {(\lambda_{2} - \lambda_{1})} ) 
    (e^{-\lambda_{2}t} - e^{-\lambda_{3}t}) 
  ] \, dt
\label{eqn:C}
\end{equation}


\begin{equation}
  \int_t^{t+\Delta t} \! D \, dt 
= \int_t^{t+\Delta t} \! 
  [ \delta e^{-\lambda_{4}t} +  
\label{eqn:D}
\end{equation}

\begin{equation} 
    \frac{\alpha \lambda_{3} \lambda_{2} \lambda_{1}}{(\lambda_{4} - \lambda_{1})(\lambda_{3} - \lambda_{1})(\lambda_{2} - \lambda_{1})} 
    (e^{-\lambda_{1}t} - e^{-\lambda_{4}t})  + \\  
\label{eqn:DA}
\end{equation}

\begin{equation}  
    \frac{\lambda_{3} \lambda_{2}} {(\lambda_{4} - \lambda_{2})(\lambda_{3} - \lambda_{2})} 
    (\beta - \frac{\alpha \lambda_{1}} {(\lambda_{2} - \lambda_{1})} ) 
    (e^{-\lambda_{2}t} - e^{-\lambda_{4}t}) + \\   
\label{eqn:DB}
\end{equation}

\begin{equation}   
    (\gamma - \frac{\alpha \lambda_{2} \lambda_{1}} {(\lambda_{3} - \lambda_{1}) (\lambda_{2} - \lambda_{1}) } - 
              \frac{\lambda_{2}} {(\lambda_{3} - \lambda_{2})} (\beta - \frac{\alpha \lambda_{1}} {(\lambda_{2} - \lambda_{1})} ) ] 
    (e^{-\lambda_{2}t} - e^{-\lambda_{4}t}) 
  ] \, dt
\label{eqn:DC}
\end{equation}


\conclusions


\begin{acknowledgements}
KZ and JF are grateful to S. Schery and S. Whittestone 
for helpful comments and discussions on radon flux and concentration measurements. 
The radon flux maps for Europe, Russia, and USA are kindly provided by 
F. Connen. This work is supported by Max Planck Society and the EUCAARI 
project.  
\end{acknowledgements}


%--- Bibliography -------------------
\bibliographystyle{copernicus}
\bibliography{bib.radon}





\end{document}
